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| Curve Sketching |
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| The graph of a given function gives a visual presentation of the behaviour of the function involving a study of symmetry, rise and fall, region of existence, passage through specific points etc. |
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| The following points are helpful to trace the curve. |
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| a) Symmetry about x - axis |
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| If the equation of the curve remains unaltered when y is replaced by -y, then the curve is symmetrical about x-axis. |
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| b) Symmetry about y-axis |
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| If the equation of the curve remains unaltered when x is replaced by -x then the curve is symmetrical about y-axis. |
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| c) Symmetry about y = x |
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| If the equation of the curve remains unaltered if x and y are interchanged, then the curve is symmetrical about y = x. |
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| d) Symmetry about y = -x |
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| If x and y are replaced by -y and -x and the equation of the curve is unaltered, then the curve is symmetrical about the line y = -x. |
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| e) Symmetry in opposite quadrants |
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| If the equation of the curve is unaltered, when x and y are replaced by -x and -y, then it is symmetrical in opposite quadrants. |
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| Example: |
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| Sketch the curve |
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| y = - sin 2x ….(1) |
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| Solution: |
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| Symmetry |
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| (a) By replacing y by -y, the equation (1) is altered, therefore the curve is not symmetrical about x-axis. |
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| (b) By replacing x by -x, the equation of the curve is altered, therefore the curve is not symmetrical about y-axis. |
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| (c) Replace x and y by -x and -y respectively in the equation y = - sin 2x, the equation of the curve remain unaltered. Therefore the curve is symmetrical in opposite quadrants. |
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| Put x = 0, y = -sin2x = 0. This implies (0, 0) is a point on the curve or the curve passes through the origin. |
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| The points of intersection with x-axis is determined by letting y = 0. |
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| Putting y = 0, - sin 2x = 0 |
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 This implies the curve intersects the x-axis at the points where  |
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| The points at which the tangent is parallel to x-axes are determined by solving. |
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| y = - sin 2x |
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 The tangent is parallel to x- axis at |
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| We know that sin x is a periodic function with 2p, that is |
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| sin(2p + x) = sinx |
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 The pattern of the curves repeats at an interval p. |
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| Therefore it is sufficient to sketch the curve form 0 to p. |
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| Determination of Few Point |
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| with this information, we sketch the graph as show . |
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